Theorem bd0 15240

Description: A formula equivalent to a bounded one is bounded. See also bd0r 15241. (Contributed by BJ, 3-Oct-2019.)

Hypotheses

Ref	Expression
bd0.min	⊢ BOUNDED 𝜑
bd0.maj	⊢ (𝜑 ↔ 𝜓)

Assertion

Ref	Expression
bd0	⊢ BOUNDED 𝜓

Proof of Theorem bd0

Step	Hyp	Ref	Expression
1		bd0.min	. 2 ⊢ BOUNDED 𝜑
2		bd0.maj	. . 3 ⊢ (𝜑 ↔ 𝜓)
3	2	ax-bd0 15229	. 2 ⊢ (BOUNDED 𝜑 → BOUNDED 𝜓)
4	1, 3	ax-mp 5	1 ⊢ BOUNDED 𝜓

Syntax hints:   ↔ wb 105  BOUNDED wbd 15228

This theorem was proved from axioms:  ax-mp 5  ax-bd0 15229

This theorem is referenced by:  bd0r  15241  bdth  15247  bdnth  15250  bdnthALT  15251  bdph  15266  bdsbc  15274  bdsnss  15289  bdcint  15293  bdeqsuc  15297  bdcriota  15299  bj-axun2  15331